Question

Ahmad and Chris each run firms that produce nails. The nails they produce are identical, and they compete in Bertrand competition. Both Ahmad and Chris are going to retire at the end of this year, so this is a one-shot game for them. Both Ahmad and Chris have a marginal cost of $1 per pound of nails. Demand in the nail market is given by P=20-Q/100, where P is the price per pound of nails, and Q is the total quantity of nails sold. Prices are discrete, set to the nearest cent, and if both Ahmad and Chris charge the same price, they will split demand evenly. Which of the following is the Nash equilibrium of this game?

Note: In the options below, P-Ahmad refers to the price that Ahmad charges, and P-Chris is the price that Chris charges.

Group of answer choices

Only (P-Ahmad=$1.01, P-Chris=$1.01)

None of the other options

Only (P-Ahmad=$1.00, P-Chris=$1.00)

This game has no Nash equilibrium

Both (P-Ahmad=$1.01, P-Chris=$1.01) and (P-Ahmad=$1.00, P-Chris=$1.00)

Answer 1

Nash Equilibrium is a situation where the involved parties in the game does not have any incentive to deviate to some other strategy

In this case, both Ahmad and Chris have equal marginal cost of $1 per pound. In a Bertrand set-up, there exists a unique pure strategy nash equilibrium where the products are homogenous and all players have equal marginal cost. This pure strategy NE is given by the following condition:

P1 = P2 = ...= Pn = MC

where P1 is the price of player 1 and so on.

In this game, there are only two players Ahmad and Chris

So, P-Ahmad = P-Chris = Marginal Cost is the Nash Equilibrium

P-Ahmad = P-Chris = $1.00 is the NE

3rd option is correct

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